Random-Packing Dynamics in Granular Flow
Martin Z. Bazant
Department of Mathematics, MIT
The Dry Fluids Laboratory @ MIT
Students:
Chris Rycroft, Ken Karmin, Jeremie Palacci,
Jaehyuk Choi (PhD ‘05)
Collaborators:
Arshad Kudrolli (Clark University, Physics)
Andrew Kadak (MIT Nuclear Engineering)
Gary Grest (Sandia National Laboratories)
Ruben Rosales (MIT Applied Math)
Support: U. S. Department of Energy,
NEC, Norbert Weiner Research Fund
Voronoi tesselation of a dense granular flow
MIT Modular Pebble-Bed Reactor
http://web.mit.edu/pebble-bed
Reactor Core :
• D = 3.5 m
• H = 8-10m
• 100,000 pebbles
• d = 6 cm
• Q =1 pebble/min
MIT Technology Review (2001)
Experiments and Simulations
MIT Dry Fluids Lab
http://math.mit.edu/dryfluids
Half-Reactor Model
(“coke bottle”)
Kadak & Bazant (2004)
Plastic or glass beads
3d Full-Scale DiscreteElement Simulations
(“molecular dynamics”)
Rycroft, Bazant, Landry, Grest (2005)
Sandia parallel code from Gary Grest
Frictional, visco-elastic spheres
N=400,000
Quasi-2d Silo Experiments
Choi et al., Phys. Rev. Lett. (2003)
Choi et al., J. Phys. A: Condens. Mat. (2005)
d=3mm glass beads
Digital-video particle tracking near wall
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
How does a dense random packing flow?
• Dilute random “packing”
(gas)
• Boltzmann’s kinetic theory:
sudden randomizing collisions
• Dense, ordered packing
(crystal)
QuickTime™ and a
Video decompressor
are needed to see this picture.
• Vacancy/interstitial diffusion
• Dislocations and other defects
• Dense, random packing
(liquid, glass, granular,...)
• Long-lasting, many-body contacts
• Are there any “defects”?
• How to describe cooperative
random motion?
Experimental Apparatus
Dry Fluids Lab (MIT Applied Math)
D=2.5cm
H=1m
L = 20cm
• t= 1ms (1000 frames/s)
vz
• Diffusion and mixing
is studied in nearly uniform flow
• W: 8, 12, 16, …, 32 mm
• vz  (W-d)1.5
d=3mm
z
• Quasi-2D silo
• W,D,L can be varied
y
x
W
• Image processing:
centeroid technique
(d=15 pixels, x~0.003d)
Universal Crossover in Diffusion
J. Choi, A. Kudrolli, R. R. Rosales, M. Z. Bazant, Phys. Rev. Lett. (2003).
Normal
Diffusion
• Super-diffusion for z << d,
Diffusion for z >> d
Super
Diffusion
Normal
Diffusion
Super
Diffusion
• x2  t :  = 1.5  1.0
• z2  t :  = 1.6  1.0
• Data collapses for all flow
rates as a function of
distance dropped (not time)
• Suggests that fluctuations
have the same physical
origin as the mean flow
(not internal “temperature”)
The Mean Velocity in Silo Drainage
Choi, Kudrolli & Bazant, J. Phys. A: Condensed Matter (2005).
Kinematic Model
u
Nedderman & Tuzun,
Powder Tech. (1979)
v v + dv
Diffusion equation…
v
v
 2v
u b
,
b
x
z
x 2
Green function = point opening
Parabolic streamlines
v ( x , z  0 )  Q ( x )
v
Q
x2
exp( 
)
4bz
4bz
…but what is diffusing?
“The Paradox of Granular Diffusion”
Particles diffuse much more slowly than free volume.
Experiment by A. Samadani & A. Kudrolli (3mm glass beads)
Microscopic Flow Mechanisms
for Dense Amorphous Materials
1. “Vacancy” mechanism for flow
in viscous liquids (Eyring, 1936);
2. “Void” model for granular drainage
(Litwiniszyn 1963, Mullins 1972)
Also: “free volume” theories of glasses
(Turnbull & Cohen 1958, Spaepen 1977,…)
3. “Spot” model for random-packing dynamics
(M. Z. Bazant, Mechanics of Materials 2005)
4. “Localized inelastic
transformations”
(Argon 1979, Bulatov & Argon 1994)
“Shear Transformation Zones”
(Falk & Langer 1998, Lemaitre 2003)
Correlations Reduce Diffusion
Simplest example: A uniform spot affects N particles.
• Volume conservation (approx.)
• Particle diffusion length
Experiment: 0.0025
DEM Simulation: 0.0024
(some spot overlap)
Direct Evidence for Spots
Spatial correlations in velocity fluctuations
(like “dynamic hetrogeneity” in glasses)
EXPERIMENTS
• MIT Dry Fluids Lab
• 3mm glass beads, slow flow (mm/sec)
• particle tracking by digital video (at wall)
• 125 frames/sec, 1024x1024 pixels
• 0.01d displacements
SIMULATIONS
• Discrete-element method (DEM), spheres
• Sandia parallel code on 32-96 processors
• Friction, Coulomb yield criterion
• Visco-elastic damping
• Hertzian or Hookean contacts
DEM
Spot Model
Void Model
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Simple spot model predicts mean flow and tracer diffusion in silo drainage
fairly well (with only 3 params), but does not enforce packing constraints.
Simulations by Chris Rycroft
“Multi-scale” Spot Algorithm
1. Simple spot-induced motion
• Apply the usual spot displacement first to all
particles within range
“Multi-scale” Spot Algorithm
2. Relaxation by soft-core repulsion
• Apply a relaxation step to all particles within a larger radius
• All overlapping pairs of particles experience a normal repulsive
displacement (soft-core elastic repulsion)
• Very simple model - no “physical” parameters, only geometry.
“Multi-scale” Spot Algorithm
3. Net cooperative displacement
• Mean displacements are mostly determined by basic spot
motion (80%), but packing constraints are also enforced
• Can this algorithm preserve reasonable random packings?
• Will it preserve the simple analytical features of the model?
Multiscale Spot
Algorithm in
Two Dimensions
• Very strong tendency for crystal order
• (Artificial) flow by dislocations,
grain boundaries, similar to crystals
• Fundamentally different from 3d…
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Multiscale Spot Model
Spot Model
In Three Dimensions
Rycroft, Bazant, Grest, Landry (2005)
 Infer 3 spot parameters
from DEM, as from expts:
* radius = 2.6 d
* volume = 0.33 v
* diffusion length = 1.39 d
 Relax particles each step
with soft-core repulsion
 “Time” = number of
drained particles
 Very similar results as
DEM, but >100x faster!
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
DEM
Spot Model vs. DEM: Mean velocity profile
Excellent fit near bottom, but upper flow is more plug-like in MD.
(Only one parameter, b, was fitted.)
Spot Model vs. DEM: Particle Diffusion
• Relaxation only slightly enhances diffusion (spot-induced rearrangements)
• Short-range (z<d) super-diffusion is not reproduced by the Spot Model.
Spot Model vs. DEM: Velocity correlations
Very similar structure of dynamical correlations.
(Only the decay length was fitted.)
Spot Model vs. DEM: “bond lengths” g(r)
t = 1.0-1.4 sec
Identical flowing structure, different from the initial condition,
after substantial drainage has occurred. (Steady state?)
Spot Model vs. DEM: “bond angles”
t = 1.0-1.4 sec
Nearly identical flowing structure, different from the initial condition.
Universal features of dense flowing hard spheres?
Universal Flowing States in Spot Simulations
1.
Taller Silos, more particles,
longer drainage times
g(r)
• Reaches statistical steady state
• Spot algorithm never breaks down
• Some local ordering compared to DEM
(involving only 1% of particles)
• Seems to “converge” to DEM state
(no ordering) as spot step size decreases
• Largely insensitive to other parameters
2. Periodic boundary conditions,
unbiased spot diffusion (“glass”)
• Also reaches statistical steady state
• “Universal” for fixed volume fraction
(independent of initial conditions,
mostly insensitive to parameters)
• New simulation method for
(nearly) hard sphere systems
DEM flowing state
Spot t = 8 sec
Spot t = 16 sec
A Mathematical Theory of
Spot-Induced Particle Motion
Bazant, Mechanics of Materials (2005)
Spot influence function
A non-local stochastic differential equation for tracer diffusion:
Interpret as the limit of a random (Riemann) sum of random variables:
Mean-Field Approximation (for spots)
Assume:
• Poisson process for spot positions with (given) mean
• Independent spot displacements
Fokker-Planck equation
Fluid velocity (opposes spot drift + climbs gradient in spot density)
Diffusivity tensor
In this way, particle flow and diffusion are expressed entirely in terms of spot dynamics,
which must be derived from mechanical principles for a given material and geometry...
Classical Mohr-Coulomb Plasticity
1. Theory of Static Stress in a Granular Material
Assume “incipient yield” everywhere:
(t/s)max = m
m= internal friction coefficient
= internal failure angle = tan-1m
2. Theory of Plastic Flow (only in a wedge hopper!)
Levy flow rule / Principle of coaxiality:
Assume equal, continuous slip
along both incipient yield planes
(stress and strain-rate tensors have same eigenvectors)
e =
p/4/2
Failures of Classical Mohr-Coulomb
Plasticity to Describe Granular Flow
1. Stresses must change from active to
passive at the onset of flow in a silo
(to preserve coaxiality).
Free surface
Silo bottom
“Active” (gravity-driven stress)
“Passive” (side-wall-driven flow)
Exit
2. Walls produce complicated velocity and stress discontinuities
(“shocks”) not seen in experiments with cohesionless grains.
3. No dynamic friction
4. No discreteness and randomness in a “continuum element”
Idea (Ken Kamrin):
Replace coaxiality with
an appropriate discrete
spot mechanism
d
Slip lines
Spot
2
Stochastic “Flow Rule”
D
Spots random walk along
Mohr-Coloumb slip lines
(but not on a lattice)
Similar ideas in lattice models for glasses:
Bulatov and Argon (1993), Garrahan & Chandler (2004)
A Simple Theory of Spot Drift
Spot = localized failure where  is replaced by k (static to dynamic friction)
(Assume quasi-static global mean stress
distribution is unaffected by spots.)
Net force on the particles
affected by a spot
A spot’s random walk is biased by this force projected along slip lines.
Use the resulting spot drift and diffusivity in the general theory
to obtain the fluid velocity and particle diffusivity…
Towards a General Theory of Dense Granular Flow?
Gravity-driven drainage from a wide quasi-2d silo: Predicts the kinematic model
Slip lines and spot drift vectors
Mean velocity profile
Coette cell with a rotating rough inner wall: Predicts shear localization
Trouble in Paradise?
Spot Model
DEM
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Particle Voronoi Volumes
Chris Rycroft, PhD thesis
• “Free volume” peaks in regions of highest shear, not the center!
• Maybe somehow alter spot dynamics, or don’t use spots everywhere…
General Theory of Dense Granular Materials:
Stochastic Elastoplasticity?
• When forcing is first applied, the granular material
responds like an elastic solid.
• Whenever the stability criterion is violated, a
plastic “liquid” region at incipient yield is born.
• Stresses in elastic solid and plastic liquid regions
evolve under load, separated by free boundaries.
• Solid regions remain stagnant, while liquid regions
flow by stochastic plasticity. Microscopic packing
dynamics is described by the Spot Model.
Classical engineering
picture of silo flow with
multiple zones, without any
quantitative theory.
(Nedderman 1991)
Conclusion
• The Spot Model is a simple, general,
and realistic mechanism for the
dynamics of dense random packings
• Random walk of spots along slip lines
yields a Stochastic Flow Rule for
continuum plasticity
• Stochastic Mohr-Coulomb plasticity
(+ nonlinear elasticity?) could be a
general model for dense granular flow
• Interactions between spots?
Extend to 3d, other materials,…?
For papers, talks, movies,… http://math.mit.edu/dryfluids
Mohr-Coulomb Stress Equations (assuming incipient yield everywhere)
Characteristics
(the slip-lines)
e
= p/4-/2
• Spot drift opposes the net material force.
• Spots drift through slip-line field constrained only by the geometry of the
slip-lines. Probability of motion along a slip-line is proportional to the
component of –Fnet in that direction. Yields drift vector:
• For diffusion coefficient D2, must solve for the unique probability
distribution over the four possible steps which generates the drift vector and
has forward and backward drift aligned with the drift direction.
D1